Hempel, C. G. (1965) Aspects of Scientific Explanation

Papineau, D. (1995) ‘Methodology: The Elements of the Philosphy of Science’ in Grayling, A. C. (ed.) Philosophy 1, pp. 123 ff.

Introduction

The Deductive-Nomological Model

Now let’s look at the things which count as the explanantia (the things which explain) in one of these Aristotelian explanations. The claim that ugly parents have ugly children is the sort of thing that we would identify as proposing a kind of Law of Nature; and it’s of a kind with laws which say that if you put a stick in water it will appear to be bent, or that if you drop a weight it will fall downwards at a constant acceleration, or that if an object emitting light is moving away from us then the light we see from it will be shifted towards the red end of the spectrum, and so on… Laws of nature of this kind, as you see, make a claim that given some initial conditions, there will be a fixed outcome which can be deduced logically from the statement of the initial conditions and the statement of the law of nature.

The most influential modern view of scientific explanations generalises this form of explanation so that the explanans is always supposed to be composed of initial conditions and applicable laws of nature. The idea is that if the explanandum can be deduced from the initial conditions and the laws of nature involved then it is explained by them. In this view all explanations look like this:

                                                C₁, C₂, C₃, …                                         the initial conditions

                                                L₁, L₂, L₃, …                                          and the relevant laws of nature

                                                --------------------                                     explain

                                                E                                                             the explanandum

In the example just given, the single initial condition would be the fact that the parent of the child is ugly, and the single applicable law of nature would be that ugly parents have ugly children.

                                                The parent is ugly                                the initial condition

                                                Ugly parents have ugly children       and the relevant law of nature

                                                ---------------------------------------            explain

                                                The child is ugly                                   the explanandum

Because it is so essentially concerned with deductions from natural laws, this model of scientific explanation is called the Deductive-Nomological model, (where ‘nomological’ just means having to do with laws,) or the Covering Law model. It was largely popularised by Carl Hempel in the mid-20^th C. He set out 4 criteria for a satisfactory scientific explanation:[1]

R1.   “The explanandum must be a logical consequence of the explanans”

R2.   “The explanans must contain general laws … required for the derivation of the explanandum”

R3.   “The explanans … must be capable … of test by experiment or observation”

R4.   “The sentences constituting the explanans must be true”

An example of the sort of thing that lends itself to this model of explanation can be seen in an explanation of why we find that pressure in a sealed container has increased to, say, 20psi after its volume has decreased. (This is what happens in a piston, if you want to imagine something real.) You have the initial conditions: that the original volume was 1 l and the original pressure was 10 kPa, and the volume was then decreased to ½ l. And we also have a relevant law of nature that relates pressure and volumes as P₁V₁ = P₂V₂. So we can explain the observation as follows:

                                                P₁ = 10 kPa                                           the initial conditions

                                                V­₁ = 1 l

                                                V­₂ = ½ l

                                                P₁V₁ = P₂V₂                                                  and the relevant law of nature

                                                ---------------------------------------            explain

                                                P₂ = 20 kPa                                           the explanandum

Here we can see that the explanandum is a logical consequence of the explanans, the explanans contains a general law that is required for the deduction (the last line of the explanans,) there are certainly tests or observations that can be made on those statements, and, the last of Hempel’s criteria, we have every reason to think that the statements in the explanans are true. So this is a good scientific explanation by Hempel’s lights.

Of course, you might think that an explanation like that one isn’t particularly explanatory. Yes, in one sense you have learned why the pressure increased – because the volume decreased; but in another sense you haven’t really gotten to the heart of things until you know why there is that relationship between pressure and volume. You need someone to explain why P₁V₁ = P₂V₂ before the explanation above can be a real explanation.

There are a couple of things to say about this objection. First, it’s a perfectly reasonable request for more information. But it is the sort of request that can be made repeatedly, for any offered explanation, until the explainer is forced to admit that no further explanation can be given. And this need not be due to the ignorance of the explainer: it’s perfectly possible that there are going to be a few brute facts about the way the universe is which are just inexplicable. It wouldn’t follow then that all of the previously offered explanations were really not explanations; but if explanations don’t have to be complete explanations in order to be real explanations then it makes no sense to say that the first explanation offered (the one in the example) is not a real explanation because it’s incomplete. It is a real explanation – one that suggests further questions for even fuller understanding.

Second, the explanation given was an explanation of a particular observed event – what is called a particular explanation – but the requested explanation is not of an observed fact but of a general truth, a natural law, or theory. Explanations of that sort can also be given in the D-N model and they are called theoretical explanations. The law in the explanandum is explained by being deducible from laws in the explanans. We could, for example, explain the relationship between pressure and volume in terms of (more fundamental?) laws involving the properties of molecular motion and collisions, but that’d be a bit tricky. In a simpler example we could explain the general truth that projectiles follow parabolic paths by noting that the vertical position of a projectile is governed by the rule y = v_0yt + ½gt² (where v_0y is the vertical component of the initial velocity) and the horizontal position of a projectile is governed by x = v_0xt (where v_0x is the horizontal component.) A little bit of maths[i] will suffice to show that

                                                y = v_0yt + ½gt²                                     relevant laws of nature

                                                x = v_0xt

                                                ---------------------------------------            explain

                                                (x, y) describes a parabola                  a law of nature   

is deductively valid. We can also agree that the laws in the explanans are required for the deduction, they are testable, and they are true; so this also satisfies Hempel’s criteria for a scientific explanation.

Testing the Model

Well, that’s Hempel’s idea of a scientific explanation; but is it satisfactory? Does it completely describe what we take to be scientific explanations? There are at least two questions that we need to ask. First, does it give sufficient conditions for something to be a scientific explanation? That is, is it true that anything that satisfies the criteria we’ve suggested should count as a scientific explanation? And second, does it give necessary conditions? Are there any things that we’d want to call scientific explanations that fail to fit the D-N model?

Sufficiency

Let’s start with the doubtful sufficiency of Hempel’s DN conditions. Here’s a common example that shows their insufficiency.

                                                The length of shadow of the flagpole is s

                                                The sun is at an angle of q

                                                When the shadow length is s and the angle of the light is q the pole length is p

                                                ---------------------------------------           

                                                The pole length is p

This is a perfectly good deduction, but we’d hardly call it an ‘explanation’ of why the pole is that length.

The problem here seems to be that we intuitively understand an explanation to be offering an effective cause (remember the Aristotelian introduction) but the statement of the nature of explanations that Hempel gives restricts itself to the formal nature of the deduction which the explanation requires. There was a good reason for this, and it wasn’t just an oversight. Hempel (and not just him) was unwilling to appeal to such notions as causality or necessity or counterfactuality in their definitions of explanations. All such ‘modal’ notions were understood to be interdefinable – causality could be defined in terms of necessity, necessity in terms of counterfactuals, counterfactuals in terms of causes, and so on – and all of them were equally suspect. It seemed to be extraordinarily difficult to get a definition of any of them that got outside this little circle. We’ve already seen, for example, how Hume made the independent ‘modal’ notion of causality a problematic concept.

We’ve also seen, however, that we aren’t bound to agree with Hume that causality is an ‘illegitimate’ concept, so it’s open to us to add a further criterion to those already listed. Something like:

R5.   “One of the laws in the explanans must describe a causal relationship”[1]

But we’d have to be a bit careful about this, because we don’t (do we?) want to exclude from science explanations like the previous explanation of the parabolic nature of trajectories of objects, which don’t seem to have causal laws involved. One solution to this that has been offered is to make the further restriction that explanations of particular events (the particular explanations that we mentioned before) have to make causal claims, but that other explanations are exempt from this restriction. If there is a way of defending this proposal from charges of ad hockery, this might solve that problem

Another less easily averted difficulty for this sort of proposal, however, is that if the definition of an explanation refers essentially to causality then we must face the further (equivalent) problem of determining the actual nature of a causal relationship. We’ll leave that question alone and move on to the second problem – that of the doubtful necessity of the DN conditions.

Necessity

Consider the explanation that could be offered to explain why little Henry has contracted mumps. You say that he contracted them from his friend Albert with whom he was playing just several hours before Albert was diagnosed as having mumps. This seems to most people to be a perfectly reasonable explanation. Your explanatory structure in DN terms would have to be something like this:

                                                Henry played with Albert

                                                Albert had mumps

                                                If you play with someone with mumps you will catch mumps

                                                ---------------------------------------------------------------------------

                                                Henry caught mumps

As an argument it’s deductively valid, there’s a law that’s necessary for the deduction, the statements are all testable, and the law describes a causal relationship. On the other hand, we don’t think that it’s true that if you play with someone you will catch mumps. We might think that it’s true that if you play with someone with mumps you’re very likely to catch the disease, but it’s not an absolute certainty. According to the (modified) DN criteria above, that would fail to be a scientific explanation.

This is disturbing because there are vast numbers of explanations that we think of as scientific, and supported by scientific data, that are in exactly the same boat as this explanation of Henry’s mumps. Just to take the most obvious example, if arguments like this one are ruled out of bounds, then we couldn’t ‘scientifically’ explain why someone who has smoked 10 packs of cigarettes a day has lung cancer.

1. Heroic Denial

There are a couple of pretty obvious solutions to this difficulty. In the first place, you could simply bite the bullet and deny that the explanation offered really is a scientific explanation. The fact that not everyone who comes in contact with a person who has mumps catches mumps indicates that’s there’s more to the catching of mumps in any individual case than the simple fact of contact with an infected person. Only when you can account for that variation, only when you know those other conditions, can you really be in a position to offer an acceptably scientific explanation of Henry’s mumps. Those additionally required initial conditions might be such things as ‘the mumps virus is transferred to the uninfected person’, ‘the infected person does not possess defensive antibodies’, ‘the mumps virus is a healthy one of its kind’, etc. In that case we might very well be able to construct an explanation on the DN model, but with a very different natural law involved.

Most people, however, don’t find this to be a very satisfactory solution. It seems to limit the types of explanations that are acceptably scientific too drastically. We must be allowed to have this sort of high level explanation, without being required to be able to dot every ‘i’ and cross every ‘t’ in the lower levels of explanation. The objection may be compared to the objection earlier to the use of high level laws relating pressure and volume.

2. Inductive-Statistical Explanations

The other pretty obvious solution, is to allow probabilistic laws in the explanans. So we could have an explanation that looks like this for Henry’s mumps

                                                Henry played with Albert                   the initial conditions

                                                Albert had mumps                             

                                                If you play with someone with         and the relevant probabilistic law

                                                mumps you are very likely

                                                to catch mumps

                                                -------------------------------------------       make likely

                                                Henry caught mumps                         the explanandum

But now we don’t have a Deductive-Nomological system, because the argument is not a deduction but an induction. And the relevant law is now not a universal regularity but only a probabilistic or statistical generalization. For this reason Hempel calls this the Inductive-Statistical model of scientific explanation (IS.)

Now remember that a good inductive argument is one that gives us reason to be confident (but not certain) that if the premisses are true then the conclusion will be true. For probabilistic inductive arguments like those we’re talking about now, this condition is only met when we judge that the premises make the conclusion very likely to be true. Therefore one of the essential conditions an IS explanation has to meet is that the probability of the explanandum is very high when the explanans is true. This condition, however, is a problem, because it still eliminates a lot of good explanations. On the IS model, we still couldn’t ‘scientifically’ explain, for example, why someone who has smoked 10 packs of cigarettes a day has lung cancer, because the incidence of lung cancer is actually pretty low in absolute terms even in heavy smokers – although it is significantly higher in smokers than in non-smokers, which is why we’re so keen to be able to make the claim that someone’s heavy smoking explains the fact of their cancer.

To Be Continued

So, the DN model of scientific explanations has some serious problems; but are they fatal? Some have thought so and so there is a mini industry in devising more satisfactory general models of explanation. For example, Wesley Salmon has proposed that we (1) lower our requirements for the relationship between the explanans and the explanandum even further – from the explanans making the explanandum highly likely to the explanans being merely statistically relevant to the explanandum, and (2) give up the idea that the explanans and the explanandum are related as the premisses and conclusion of an argument. For Salmon, an explanation is just a collection of facts about an event that are statistically relevant and causally significant. From another direction entirely, Bas van Fraassen has proposed that we take explanations to be certain sorts of answers to questions ‘why’ something is the case rather than some other things being the case, where the type of answer expected is sensitive to the context in which the question is asked. This is the ‘pragmatic’ model of explanations.

All of these offer different answers to the question of what is an explanation, and they all have their own difficulties. By looking relatively closely at just the DN model we’ve been able to identify some of the sorts of difficulties that it’s likely any model of explanation is going to face. We can do no more now.